Finite nonsuspension spectra with metastable structures

Construct an explicit finite spectrum that is not a suspension spectrum but nevertheless admits a metastable structure.

Background

The paper studies spectra equipped with a metastable structure, defined as a lift of the Tate diagonal from the spectrum to the homotopy fixed points of its tensor square. Suspension spectra provide the principal examples. A dimension-connectivity criterion shows that sufficiently connective finite spectra admit metastable structures, but the authors note that this criterion does not produce examples beyond suspension spectra.

The unresolved problem is to find a genuinely nonsuspension finite spectrum that still admits such a structure, which would clarify how far metastable structures extend beyond those induced by space-level diagonals.

References

Can one give an explicit example of a finite spectrum which is not a suspension spectrum but nevertheless admits a metastable structure?

— Metastable homotopy theory via Tate coalgebras  (2609.31274 - Nervo, 25 Sep 2026) in Section 1, subsection “Definition and examples,” immediately after Remark following Lemma 1.10

We do not know whether this can be strengthened to show that \pi_0X cannot contain any 2-torsion. In particular, it is not known whether the Moore spectra \mathbb{S}/2n admit a metastable structure for any n > 1. More generally, we do not know whether \mathbb{S}/\alpha admits a metastable structure for any nonzero, noninvertible 2-local map \alpha.

— Metastable homotopy theory via Tate coalgebras  (2609.31274 - Nervo, 25 Sep 2026) in Section 1, subsection “Obstructions,” Remark immediately preceding Question